↑

Properties

Label 2-2523-87.80-c0-0-2
Degree $2$
Conductor $2523$
Sign $0.620 + 0.784i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.900 + 0.433i)3-s + (0.222 − 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.623 − 0.781i)11-s + (−0.623 + 0.781i)13-s + (0.222 − 0.974i)14-s + (−0.623 + 0.781i)16-s + 17-s + (0.623 − 0.781i)18-s + (0.623 + 0.781i)21-s + (−0.900 − 0.433i)22-s + (−0.222 − 0.974i)24-s + (−0.900 + 0.433i)25-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.900 + 0.433i)3-s + (0.222 − 0.974i)6-s + (0.900 + 0.433i)7-s + (−0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (0.623 − 0.781i)11-s + (−0.623 + 0.781i)13-s + (0.222 − 0.974i)14-s + (−0.623 + 0.781i)16-s + 17-s + (0.623 − 0.781i)18-s + (0.623 + 0.781i)21-s + (−0.900 − 0.433i)22-s + (−0.222 − 0.974i)24-s + (−0.900 + 0.433i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.620 + 0.784i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1037, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.620 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.779894756\)
\(L(\frac12)\) \(\approx\) \(1.779894756\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 \)
good2 \( 1 + (0.222 + 0.974i)T + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.900 - 0.433i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + (-0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + 2T + T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.623 - 0.781i)T^{2} \)
67 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.900 + 0.433i)T^{2} \)
79 \( 1 + (0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T + (-0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.623 + 0.781i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.062984084818235692144053012169, −8.529666158509491563763641689339, −7.67810613725052889893878550109, −6.83235317800709142146737978998, −5.74942279839492074761531418050, −4.83393174029117698606661653496, −3.80782940858434823116273448153, −3.19619537241012289496920419142, −2.12464774612047355958199990126, −1.50422041062164268881109885976, 1.46367105230593812241287451839, 2.45922063554629742235526816478, 3.48980652330490030812847950301, 4.54058348087856805188807201916, 5.44804104239928780293585493216, 6.40868037634175113343799721545, 7.18934936123385080128043644076, 7.69820461110891227431018528694, 8.115360628252380384412767805033, 8.895257649294070106431496744333

Graph of the $Z$-function along the critical line